Rotational motion, Gravitation, Simple harmonic motion

Rotational Motion

A rotation is a circular movement of an object around a center (or point) of rotation . A three-dimensional object always rotates around an imaginary line called a rotation axis. If the axis passes through the body’s center of mass, the body is said to rotate upon itself, or spin.

A rigid body is a body that can rotate with all the parts locked together and without any change in its shape.

The moment of inertia of a rigid body is given by

I = M K² where M is the total mass of the body and K is called the radius of gyration of the body. The radius of gyration is that distance from the axis of rotation where the whole mass of the body can be assumed to be placed to get the same moment of inertia which the body actually has.Few common Moment of Inertia are given below:-

Rotational Equations governing Rotational Motion are:-

Gravitation

Gravitation is the force by which every mass or particle of matter, including photons, attracts and is attracted by every other mass or particle of matter.The gravitational field at any point P in space is defined as the gravitational force felt by a tiny unit mass placed at P.

Every particle of matter in the universe attracts every other particle with a force that is directly proportional to the product of the masses of the particles and inversely proportional to the square of the distance between them.

Mathematically, this law, and the magnitude of the force due to the gravitational interaction between two particles, is expressed with :-

Properties of Gravitational Force.

 It is always attractive in nature while electric and magnetic force can be attractive or repulsive.

 It is independent of the medium between the particles while electric and magnetic force depend on the nature of the medium between the particles.

 It holds good over a wide range of distances. It is found true for interplanetary to inter atomic distances.

 It is a central force i.e. acts along the line joining the centres of two interacting bodies.

 It is a two-body interaction i.e. gravitational force between two particles is independent of the presence or absence of other particles.

 It is the weakest force in nature : As Fnuclear > F electromagnetic > F gravitational .

 It is a conservative force i.e. work done by it is path independent or work done in moving a particle round a closed path under the action of gravitational force is zero.

 It is an action reaction pair i.e. the force with which one body (say earth) attracts the second body (say moon) is equal to the force with which moon attracts the earth.

Simple Harmonic Motion

Simple Harmonic Motion is a type of periodic motion or oscillation motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement .

During simple harmonic motion the acceleration is proportional to the displacement x and is oppositely directed. If the displacement is to the right of the equilibrium position, then the acceleration is to the left, and vice versa. The angular speed ? is a constant, a characteristic of the motion. The angular speed can be expressed in terms of the frequency, or the period, of the motion.

Characterstics of SHM:-

 It is a motion along a straight line.

 The body moves back and forth with respect to a mean position.

 The body returns to a given point in the path with the same velocity after regular intervals of time.

 The acceleration is directly proportional to its displacement from the mean position and is always directed towards it.

 The force acting on the body always tries to bring it back to its equilibrium position.

 The maximum displacements on either side of the equilibrium position are equal.

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